Question

Find principal argument of `(1 + i sqrt(3))^2`.

Answer 1

Given that: `(1 + i sqrt(3))^2 = 1 + i^2 . 3 + 2sqrt(3) i`

= `1 - 3 + 2sqrt(3)i`

= `-2 + 2sqrt(3)i`

`tan alpha = |(2sqrt(3))/2|`  ......`[because tan alpha = |("Img"(z))/("Re"(z))|]`

⇒ `tan alpha = |- sqrt(3)| = sqrt(3)`

⇒ `tan alpha = tan  pi/3`

∴ `alpha = pi/3`

Now Re(z) < 0 and image(z) > 0.

∴ arg(z) = `pi - alpha`

= `pi - pi/3`

= `(2pi)/3`

Hence, the principal arg = `(2pi)/3`.